On the optimal constants in Korns and geometric rigidity estimates, in bounded and unbounded domains, under Neumann boundary conditions

We are concerned with the optimal constants: in the Korn inequality under tangential boundary conditions on bounded sets $Omega subset mathbb{R}^n$, and in the geometric rigidity estimate on the whole $mathbb{R}^2$. We prove that the latter constant equals $sqrt{2}$, and we discuss the relation of the former constants with the optimal Korns constants under Dirichlet boundary conditions, and in the whole $mathbb{R}^n$, which are well known to equal $sqrt{2}$. We also discuss the attainability of these constants and the structure of deformations/displacement fields in the optimal sets.